Question

Factorize the following expression
\[{{p}^{2}}-{{q}^{2}}\]

Answer 1

Hint: To factorize any expression it is required to evaluate the given expression. The evaluation of the expression can be done either by using step by step process or by using the formula which is similar to the given expression. Given expression is a quadratic one. hence the maximum number of factors can be two.

Complete step by step answer:
Given expression which is quadratic is
\[{{p}^{2}}-{{q}^{2}}\]
Let us consider the expression equal to a variable \[x\] . it is given by
\[x={{p}^{2}}-{{q}^{2}}\]
Now we have to evaluate the expression as below
Let us add and subtract the term \[pq\]
The expression would become
\[x={{p}^{2}}-{{q}^{2}}-pq+pq\]
Rearrange the terms, then we have
\[\Rightarrow x={{p}^{2}}+pq-{{q}^{2}}-pq\]
Take \[p\] common from the first two terms and \[q\] common from last two terms
We get
\[\Rightarrow x=p\left( p+q \right)-q\left( q+p \right)\]
We can observe that \[\left( p+q \right)\] is common from both terms. Hence take it as a common factor t\[\Rightarrow x=\left( p+q \right)\left( p-q \right)\]
Hence the factors of the expression \[{{p}^{2}}-{{q}^{2}}\] are given by \[\left( p+q \right)\] and \[\left( p-q \right)\]

Note: The factorization of any expression means expressing the equation as the factors which are multiplied to give the following expression. For a quadratic expression the maximum number of factors are two. Similarly for cubic polynomials it will be three. The number of factors depends on the degree of the expression given.
We can also solve this expression directly by using one of the basic algebraic formula \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\] . The given expression is in the form of RHS of the formula. Hence, we can substitute the values given by \[a=p,b=q\] in the LHS side to get the required factors.